This investigation is designed to find the net rate of decrease in the component of velocity parallel to the original direction of motion of a proton moving through an electron gas exhibiting a non-spherical velocity distribution.

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In the past few decades, brake pad-rotor interface friction studies have gained high importance in the automotive industry. The goal of these studies has been to improve the design to maximize the contact area and performance in brakes. In these studies, friction coefficient has always assumed to be the same for linear and rotational motion. In our study, we show that the rotational and linear friction process have different friction coefficients. We use semi-metallic and ceramic brake material pads reduced into brake samples using scaling laws of physics. The samples were mounted on the Universal Mechanical Tester and experimented for linear and rotational friction process against Pearlitic Gray cast iron rotor. From results, it proved friction coefficients of linear movement is always higher than the rotational movement. The linear friction coefficient was found to be 43% higher on an average than the rotational friction coefficient in both the materials tested at 1MPa and 10 mm/s. These results will help industry in gaining better fundamental understanding about the friction coefficients of rotor- brake contact interfaces.

Este trabalho usa o método de determinação do coeficiente de atrito com a utilização das equações de Panknin e do ensaio de Dobramento sob Tensão (DST) para determinar o coeficiente de atrito pelas equações de Panknin foi determinada a força máxima de estampagem de copos cilíndricos dos três materiais, aço inox austenítico AISI 304, aço inox ferrítico AISI 430 e aço de baixo teor de carbono EEP, com uso de quatro lubrificantes. O valor da força máxima de estampagem foi substituída nas equações de Panknin e calculados os coeficientes de atrito. Para o ensaio de Dobramento sob Tensão foram estudadas várias equações que serviram de referência para o cálculo do coeficiente de atrito: Equação das Polias, Equação das Polias Sem Dobramento, Equação das Polias Sem Dobramento considerando os Fatores Geométricos (raio do pino e espessura da chapa), Equação de Wilson, Equação de Sniekers e Equação de Andreasen. Essas equações foram utilizadas para determinar o coeficiente de atrito com a utilização de dois lubrificantes. Os materiais de estudo foram o aço de baixo teor de carbono (EEP), aço inoxidável austenítico (AISI 304) e aço inoxidável ferrítico (AISI 430). A escolha desses materiais foi justificada pela grande quantidade de produtos estampados com esses materiais. No ensaio de DST foram utilizados corpos de prova cortados a 0°, 45° e 90° em relação à direção de laminação da chapa metálica. Nos ensaios de Dobramento sob Tensão utilizou-se uma pressão constante e uma velocidade também constante. Os resultados do coeficiente de atrito calculados pelas equações do ensaio de DST foram comparados com os resultados dos coeficientes de atrito calculados pelas equações de Panknin para validação ou não do uso das equações de Panknin para determinar o coeficiente de atrito e de referência do coeficiente de atrito destes materiais pelo Ensaio de Dobramento Sob Tensão. / This paper uses two methods for calculation of the coefficient of friction efficient, using Panknin equations and the Bending under Tension test (DST). To determine the coefficient of friction with the Panknin equations, maximum stamping strength on cylindrical cups of three materials using four lubricants was calculated. The value of maximum stamping strength was replaced in the Panknin equations to calculate the fiction coefficient. For the Bending under Tension test various equations were studied which served as reference to calculate the friction coefficient: Equation of Sheaves, Equation of Sheaves without Bending, Equation of Sheaves without Bending considering the geometric factors (radius of the pin and plate thickness), Wilson Equation, Sniekers Equation and Andreasen Equation. These equations were used to determine the friction coefficient with the use of two lubricants. The studied materials were low carbon steel, austenitic stainless steel (AISI 304) and ferritic stainless steel (AISI 430). The choice of these materials was justified by the large amount of stamped products with such materials. The Bending under Tension test was performed with specimens cut at 0°, 45° and 90° to the rolling direction of the metal sheet. The tests were made under constant pressure and constant speed. The results of friction the coefficient calculated by equations of the DST test were compared to the results of the coefficients of friction calculated by Panknin equations.

Este trabalho usa o método de determinação do coeficiente de atrito com a utilização das equações de Panknin e do ensaio de Dobramento sob Tensão (DST) para determinar o coeficiente de atrito pelas equações de Panknin foi determinada a força máxima de estampagem de copos cilíndricos dos três materiais, aço inox austenítico AISI 304, aço inox ferrítico AISI 430 e aço de baixo teor de carbono EEP, com uso de quatro lubrificantes. O valor da força máxima de estampagem foi substituída nas equações de Panknin e calculados os coeficientes de atrito. Para o ensaio de Dobramento sob Tensão foram estudadas várias equações que serviram de referência para o cálculo do coeficiente de atrito: Equação das Polias, Equação das Polias Sem Dobramento, Equação das Polias Sem Dobramento considerando os Fatores Geométricos (raio do pino e espessura da chapa), Equação de Wilson, Equação de Sniekers e Equação de Andreasen. Essas equações foram utilizadas para determinar o coeficiente de atrito com a utilização de dois lubrificantes. Os materiais de estudo foram o aço de baixo teor de carbono (EEP), aço inoxidável austenítico (AISI 304) e aço inoxidável ferrítico (AISI 430). A escolha desses materiais foi justificada pela grande quantidade de produtos estampados com esses materiais. No ensaio de DST foram utilizados corpos de prova cortados a 0°, 45° e 90° em relação à direção de laminação da chapa metálica. Nos ensaios de Dobramento sob Tensão utilizou-se uma pressão constante e uma velocidade também constante. Os resultados do coeficiente de atrito calculados pelas equações do ensaio de DST foram comparados com os resultados dos coeficientes de atrito calculados pelas equações de Panknin para validação ou não do uso das equações de Panknin para determinar o coeficiente de atrito e de referência do coeficiente de atrito destes materiais pelo Ensaio de Dobramento Sob Tensão. / This paper uses two methods for calculation of the coefficient of friction efficient, using Panknin equations and the Bending under Tension test (DST). To determine the coefficient of friction with the Panknin equations, maximum stamping strength on cylindrical cups of three materials using four lubricants was calculated. The value of maximum stamping strength was replaced in the Panknin equations to calculate the fiction coefficient. For the Bending under Tension test various equations were studied which served as reference to calculate the friction coefficient: Equation of Sheaves, Equation of Sheaves without Bending, Equation of Sheaves without Bending considering the geometric factors (radius of the pin and plate thickness), Wilson Equation, Sniekers Equation and Andreasen Equation. These equations were used to determine the friction coefficient with the use of two lubricants. The studied materials were low carbon steel, austenitic stainless steel (AISI 304) and ferritic stainless steel (AISI 430). The choice of these materials was justified by the large amount of stamped products with such materials. The Bending under Tension test was performed with specimens cut at 0°, 45° and 90° to the rolling direction of the metal sheet. The tests were made under constant pressure and constant speed. The results of friction the coefficient calculated by equations of the DST test were compared to the results of the coefficients of friction calculated by Panknin equations.

Este trabalho usa o método de determinação do coeficiente de atrito com a utilização das equações de Panknin e do ensaio de Dobramento sob Tensão (DST) para determinar o coeficiente de atrito pelas equações de Panknin foi determinada a força máxima de estampagem de copos cilíndricos dos três materiais, aço inox austenítico AISI 304, aço inox ferrítico AISI 430 e aço de baixo teor de carbono EEP, com uso de quatro lubrificantes. O valor da força máxima de estampagem foi substituída nas equações de Panknin e calculados os coeficientes de atrito. Para o ensaio de Dobramento sob Tensão foram estudadas várias equações que serviram de referência para o cálculo do coeficiente de atrito: Equação das Polias, Equação das Polias Sem Dobramento, Equação das Polias Sem Dobramento considerando os Fatores Geométricos (raio do pino e espessura da chapa), Equação de Wilson, Equação de Sniekers e Equação de Andreasen. Essas equações foram utilizadas para determinar o coeficiente de atrito com a utilização de dois lubrificantes. Os materiais de estudo foram o aço de baixo teor de carbono (EEP), aço inoxidável austenítico (AISI 304) e aço inoxidável ferrítico (AISI 430). A escolha desses materiais foi justificada pela grande quantidade de produtos estampados com esses materiais. No ensaio de DST foram utilizados corpos de prova cortados a 0°, 45° e 90° em relação à direção de laminação da chapa metálica. Nos ensaios de Dobramento sob Tensão utilizou-se uma pressão constante e uma velocidade também constante. Os resultados do coeficiente de atrito calculados pelas equações do ensaio de DST foram comparados com os resultados dos coeficientes de atrito calculados pelas equações de Panknin para validação ou não do uso das equações de Panknin para determinar o coeficiente de atrito e de referência do coeficiente de atrito destes materiais pelo Ensaio de Dobramento Sob Tensão. / This paper uses two methods for calculation of the coefficient of friction efficient, using Panknin equations and the Bending under Tension test (DST). To determine the coefficient of friction with the Panknin equations, maximum stamping strength on cylindrical cups of three materials using four lubricants was calculated. The value of maximum stamping strength was replaced in the Panknin equations to calculate the fiction coefficient. For the Bending under Tension test various equations were studied which served as reference to calculate the friction coefficient: Equation of Sheaves, Equation of Sheaves without Bending, Equation of Sheaves without Bending considering the geometric factors (radius of the pin and plate thickness), Wilson Equation, Sniekers Equation and Andreasen Equation. These equations were used to determine the friction coefficient with the use of two lubricants. The studied materials were low carbon steel, austenitic stainless steel (AISI 304) and ferritic stainless steel (AISI 430). The choice of these materials was justified by the large amount of stamped products with such materials. The Bending under Tension test was performed with specimens cut at 0°, 45° and 90° to the rolling direction of the metal sheet. The tests were made under constant pressure and constant speed. The results of friction the coefficient calculated by equations of the DST test were compared to the results of the coefficients of friction calculated by Panknin equations.

In this thesis we study a variety of problems in superfluid turbulence, princi-pally in two dimensions. A summary of the main results of our studies is given below; we indicate the Chapters in which we present these. In Chapter 1, we provide an overview of several problems in superfluid turbulence with special emphasis on background material for the problems we study in this thesis. In particular, we give: (a) a brief introduction of fluid turbulence; (b) an overview of superfluidity and the phenomenological two-fluid model; (c) a brief overview of experiments on superfluid turbulence; (d) an introductory accounts of the phenomenological models used in the study of superfluid turbulence. We end with a summary of the problems we study in subsequent Chapters of this thesis. In Chapter 2, we present a systematic, direct numerical simulation of the two-dimensional, Fourier-truncated, Gross-Pitaevskii equation to study the turbulent evolutions of its solutions for a variety of initial conditions and a wide range of parameters. We find that the time evolution of this system can be classified into four regimes with qualitatively different statistical properties. First, there are transients that depend on the initial conditions. In the second regime, power- law scaling regions, in the energy and the occupation-number spectra, appear and start to develop; the exponents of these power laws and the extents of the scaling regions change with time and depend on the initial condition. In the third regime, the spectra drop rapidly for modes with wave numbers k > kc and partial thermalization takes place for modes with k < kc ; the self-truncation wave number kc(t) depends on the initial conditions and it grows either as a power of t or as log t. Finally, in the fourth regime, complete thermalization is achieved and, if we account for finite-size effects carefully, correlation functions and spectra are consistent with their nontrivial Berezinskii-Kosterlitz-Thouless forms. Our work is a natural generalization of recent studies of thermalization in the Euler and other hydrodynamical equations; it combines ideas from fluid dynamics and turbulence, on the one hand, and equilibrium and nonequilibrium statistical mechanics on the other. In Chapter 3, we present the first calculation of the mutual-friction coefficients α and α (which are parameters in the Hall-Vinen-Bekharevich-Khalatnikov two-fluid model that we study in chapter 5) as a function of temperature in a homogeneous Bose gas in two-dimensions by using the Galerkin-truncated Gross-Pitaevskii equation, with very special initial conditions, which we obtain by using the advective, real, Ginzburg-Landau equation (ARGLE) and an equilibration procedure that uses a stochastic Ginzburg-Landau equation (SGLE). We also calculate the normal-fluid density as a function of temperature. In Chapter 4, we elucidate the interplay of particles and fields in superfluids, in both simple and turbulent flows. We carry out extensive direct numerical simulations (DNSs) of this interplay for the two-dimensional (2D) Gross-Pitaevskii (GP) equation. We obtain the following results: (1) the motion of a particle can be chaotic even if the superfluid shows no sign of turbulence; (2) vortex motion depends sensitively on particle charateristics; (3) there is an effective, superfluid-mediated, attractive interaction between particles; (4) we introduce a short-range repulsion between particles, with range rSR, and study two- and many-particle collisions; in the case of two-particle, head-on collisions, we find that, at low values of rSR, the particle collisions are inelastic with coefficient of restitution e = 0; and, as we in-crease rSR, e becomes nonzero at a critical point, and finally attains values close to 1; (5) assemblies of particles and vortices show rich, turbulent, spatio-temporal evolution. In Chapter 5, we present results from our direct numerical simulations (DNSs) of the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) two-fluid model in two dimensions. We have designed these DNSs to study the statistical properties of inverse and forward cascades in the HVBK model. We obtain several interesting results that have not been anticipated hitherto: (1) Both normal-fluid and superfluid energy spectra, En(k) and Es(k), respectively, show inverse- and forward-cascade regimes; the former is characterized by a power law Es(k) En(k) kα whose exponent is consistent with α 5/3. (2) The forward-cascade power law depends on (a) the friction coefficient, as in 2D fluid turbulence, and, in addition, on (b) the coefficient B of mutual friction, which couples normal and superfluid compo-nents. (3) As B increases, the normal and superfluid velocities, un and us, re-spectively, get locked to each other, and, therefore, Es(k) En(k), especially in the inverse-cascade regime. (4) We quantify this locking tendency by calculating the probability distribution functions (PDFs) P(cos(θ)) and P(γ), where the angle θ ≡ (un • us)/( |un||us|) and the amplitude ratio γ = |un|/|us |; the former has a peak at cos(θ) = 1; and the latter exhibits a peak at γ = 1 and power-law tails on both sides of this peak. (4) This locking increases as we increase B, but the power-law exponents for the tails of P(γ) are universal, in so far as they do not depend on B, ρn/ρ, and the details of the energy-injection method. (5) We characterize the energy and enstrophy cascades by computing the energy and enstrophy fluxes and the mutual-friction transfer functions for all wave-number scales k. In Chapter 6, we examine the multiscaling of structure functions in three-dimensional superfluid turbulence by using a shell-model for the three-dimensional HVBK equations. Our HVBK shell model is based on the GOY shell model. In particular, we examine the dependence of multiscaling on the normal-fluid fraction and the mutual-friction coefficients. We hope our in silico studies of 2D and 3D superfluid turbulence will stimulate new experimental, numerical, and theoretical studies.

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